Fast method for identifying coil misalignment/mutualcoupling in wireless charging systems

ABSTRACT

Methods and apparatus for determining the misalignment and mutual coupling between the transmitter coil and receiver coil, with or without an intermediate relay resonator coil, of a wireless power charging system are provided. The determination can be made without using any direct measurement from the receiver circuit. The technic involves exciting the transmitter coil of the wireless power charging system at several frequencies with equal or different input voltage/current, such that the number of equivalent circuit equations is at least equal to the number of unknown terms in the equations. The methods use the knowledge of only the input voltage and the input current of the transmitter coil. This means that the mutual inductance or magnetic coupling coefficient between the transmitter and receiver coils can be determined based on the information obtained from the transmitter circuit and there is no need for any wireless communication from or direct measurements of the receiver circuit.

FIELD OF THE INVENTION

The present invention relates generally to a fast method for identifyingcoil misalignment and mutual coupling of transmitter and receiver coilsin wireless charging systems without using direct measurement from thereceiver coil.

BACKGROUND OF THE INVENTION

While wireless power was pioneered by Nicola Tesla a century ago, it hasgained substantial attention for commercial applications only in thelast two decades. One reason for this is the emerging availability ofpower electronics, which provide a technical means to process electricalpower at high speed with high frequency switching, thus acting as anenabling technology for wireless power transfer. Generally, a wirelesspower transfer system consists of at least a transmitter module(comprising a transmitter circuit and a transmitter coil) and at leastone receiver module (comprising a receiver coil and a receiver circuit).Tesla pointed out in his research that using magnetic coupling withresonance could achieve optimal efficiency. This magnetic resonanceconcept, originated by Tesla, has been adopted by Prof. John Boys in hisinductive power transfer research for mobile robotics (since 1990's) andby Prof. Ron Hui in his planar wireless charging platforms for portableelectronic products (since 2000's). The world's first wireless powerstandard “Qi”, launched by the Wireless Power Consortium, hasincorporated the near-field magnetic coupling and resonant circuits intothe basic technology for wireless charging of a range of products.

In recent research, one major topic is related to the misalignmentbetween the transmitter coil and the receiver coil. This issue isparticularly relevant to wireless charging of electric vehicles,although it must be emphasized that this issue is relevant to many otherwireless charging applications. For portable electronics devices, suchas mobile phones, the Qi standard adopts the multilayer coil arraystructures proposed by Prof. Ron Hui so that the targeted devices (withtheir received coils embedded) can be placed anywhere on the chargingsurface. Such a feature is called “free-positioning.” But for electricvehicles, the transmitter coil is usually laid under the floor, whilethe receiver coil is located inside the vehicle. Since it is not easy topark the vehicle in the exact location each time, the misalignmentbetween the transmitter and receiver coils has become a challengingissue. A driver can hardly align the two coils visually when one of themis under ground. Some cases alignment is also limited by the parkingposition.

Some researchers have proposed the use of a wireless communicationsystem to provide feedback information from the receiver circuit to thetransmitter circuit in order to optimize the alignment, and hence thecontrol. Such a solution is reported in Onar et al., “Oak Ridge NationalLaboratory Wireless Power Transfer Development for Sustainable CampusInitiative”, 2013 IEEE Transportation Electrification Conference andExpo (ITEC), 2013, pp. 1-8 and N.Y. Kim et al., “Adaptive frequency withpower-level tracking system for efficient magnetic resonance wirelesspower transfer”, Electron. Lett., Vol. 48, No. 8, pp. 452-454, 12 April2012. However, the additional wireless communication module increasesthe cost and the overall complexity of the system.

The present inventors have previously pioneered a technique that canmonitor the output conditions of the receiver circuit without using anywireless or wired communication circuit for feeding the outputinformation back to the controller on the transmitter side. Thistechnique is disclosed in S. Y. R. Hui, D. Lin, J. Yin and C. K. Lee,“Methods for Parameter Identification, Load Monitoring and Output PowerControl in Wireless Power Transfer Systems”, U.S. Provisional PatentApplication 61/862,627 filed Aug. 6, 2013 (the ‘627 application“); andPCT patent application PCT/CN2014/083775 filed Aug. 6, 2014 (“Hui '775PCT application”), which applications are incorporated herein byreference in their entirety. In the Hui '775 PCT application there isproposed a methodology for (1) identifying the system parameters, (2)monitoring the load conditions and (3) generating output control basedon the use of the input voltage and input current in the transmittercircuit. The methodology of the Hui '775 PCT application does not useany direct measurement of the receiver circuit or the load on thereceiver side. This technology involves 2 main processes. The firstprocess uses an intelligent or evolutionary algorithm, such as a geneticalgorithm or swarm particle algorithm or their variants, to determinethe system parameters. After this first process has been completed andthe parameters have been identified, a system model can be developed. Inthe second process, the system matrix equations with the known systemparameters obtained from the first process (except the load impedance)can be re-arranged and used to calculate the load impedance with onemeasurement of the input voltage and input current. As a result, all ofthe required information, such as load power, output voltage and outputpower of the receiver circuit, etc., can be calculated. A block diagramof the concept is shown in FIG. 1. In this previous patent application,the first process usually takes a fairly long time (in terms of tens ofminutes if a fast processor is used) to determine the system parametersincluding the mutual inductance terms among the coupled windings.

It would be advantageous if the method of the Hui '775 PCT applicationcould be improved so that the calculation time could be significantlyshortened.

SUMMARY OF THE INVENTION

The present invention relates to a method and apparatus to calculatecoil misalignment in wireless charging, in which the time taken tocalculate the necessary system parameters (i.e. mutual inductancebetween coils) is greatly reduced, e.g., to less than a minute and, moreparticularly, relates to rapid calculation for coil misalignment inwireless charging of electric vehicles.

The proposed method can be implemented in the same circuit structure asshown in FIG. 1; but, the use of intelligent or evolutionary algorithmsin the first process of system parameter identification can beeliminated. Unlike the use of evolutionary algorithms, which typicallyrequire tens of minutes to determine the system parameters, thisproposed method can determine the mutual inductance terms within a veryshort time (in terms of seconds rather than tens of minutes).

In this invention, the method of identifying the system parameters forelectrical vehicle wireless charging is proposed. Since theuncertainties of the (1) coil misalignment and (2) coil distance arerelated to the mutual inductance values of the mutual coupled coils onlyand are independent of the capacitance in the coil resonators, thisinvention focuses on a method of estimating these mutual inductanceterms in a rapid manner. The assumption is that the self-inductanceterms and the resonant-capacitance terms are known from themanufacturers. Only the mutual inductance terms and the load impedanceterms are unknown.

For parking (EV charging application), since only the mutual inductancewill be changed, the proposed new method can generate sufficient numberof equations to solve the unknown system parameters very quickly (inseconds).

The current invention performs the following steps to obtain theparameters at the transmitter side for providing the best efficiency tothe receiver coil:

(1) Excite the power system at a number of different frequencies to takemore measurements for a sufficient number of equations in order to solvefor the unknown values.

(2) Solve the system matrix equation using the generated equations forthe unknown system parameter (i.e. mutual inductance value).

When the number of equations is equal to or more than the number ofunknown parameters, the equations can be solved mathematically for theunknown parameters. If constant power is required for electric vehicles(EV), the wireless power transfer (WPT) system can use different voltagemagnitudes instead of different excitation frequencies to generate therequired number of equations. Once the system parameters are identified,the WPT system can choose the proper operating frequency and inputvoltage/current to operate at its best efficiency using the same methoddisclosed in the Hui '775 PCT application.

BRIEF DESCRIPTION OF THE DRAWINGS

The present invention can be more fully understood by the followingdetailed description of the preferred embodiments, with reference madeto the accompanying drawings, wherein:

FIG. 1 is a block diagram of a wireless power transfer system withunknown system parameters which does not use feedback from the receiverfor control purposes;

FIG. 2 is a block diagram of a prior art 2-coil wireless power transfersystem;

FIG. 3 is a block diagram of a prior art 3-coil wireless power transfersystem;

FIG. 4 is a graph of a typical battery charging profiles;

FIG. 5 is block diagram of a 2-coil EV wireless charging system withac/dc and dc/dc converters according to the present invention;

FIG. 6 is a block diagram of a 3-coil EV wireless charging system withac/dc and dc/dc converters according to the present invention;

FIG. 7 is a block diagram of an n-coil wireless power transfer systemaccording to the present invention;

FIG. 8 is a block diagram of a 2-coil EV wireless charging system withac/dc and dc/dc converters, and with unknown system parameters which,according to the present invention, does not use feedback from thereceiver for control purposes;

FIG. 9 is a block diagram of a 3-coil EV wireless charging system withac/dc and dc/dc converters, and with unknown system parameters which,according to the present invention, does not use feedback from thereceiver for control purposes.

FIG. 10 is a block diagram of a 2-coil EV wireless charging system withac/dc and dc/dc converters, maximum energy efficiency tracking and withunknown system parameters determination which, according to the presentinvention, does not use feedback from the receiver for control purposes

FIG. 11 shows graphs of the output power and efficiency versus the loadresistance for the circuit of FIG. 10; and

FIG. 12 shows graphs of the input power and efficiency versus the inputvoltage for the circuit of FIG. 10.

DETAILED DESCRIPTION OF EXEMPLARY EMBODIMENTS OF THE INVENTION

The proposed method of the present invention may be explained with theuse of a 2-coil wireless power transfer system as shown in FIG. 2 and a3-coil system which has a relay resonator between the transmitter coiland the receiver coil as shown in FIG. 3. The advantage of using a3-coil system can be found in reference: W. X. Zhong et al., “AMethodology for Making a Three-Coil Wireless Power Transfer System MoreEnergy Efficient Than a Two-Coil Counterpart for Extended TransferDistance”, IEEE Transactions on Power Electronics, vol. 30, pp. 933-942,2015 (the “Zhong 2015 article 1”).

The descriptions given below are based on wireless charging of electricvehicles (EV). However, it should be noted that the proposed method isalso applicable to other wireless charging systems.

There are numerous publications on wireless charging of electricvehicles. Examples are: J. G. Hayes et al., “Wide-load-range resonantconverter supplying the SAE J-1773 electric vehicle inductive charginginterface”, Industry Applications, IEEE Transactions on, vol. 35, pp.884-895, 1999; W. Chwei-Sen et al., “Design considerations for acontactless electric vehicle battery charger”, Industrial Electronics,IEEE Transactions on, vol. 52, pp. 1308-1314, 2005; and U. K. Madawalaet al., “A Bidirectional Inductive Power Interface for Electric Vehiclesin V2G Systems”, Industrial Electronics, IEEE Transactions on, vol. 58,pp. 4789-4796, 2011. In these reports, all the parameters of thetransmitter and the receivers must be known. Such parameters include theself-inductances and the resistances of the transmitter coil(s) and thereceiver coil, the mutual inductances between all mutually coupledcoils, the capacitances connected in the transmitter circuit and thereceiver circuit, and the equivalent load impedance. Based on theseparameters, one can use circuit theory to calculate power, efficiency ofthe system, and can decide upon the charging strategy according to thetype of the battery and the condition of the battery. For EV batterycharging, the charging process usually consists of two main stages:namely (i) a constant-current (CC) charging stage and (ii) aconstant-voltage (CV) charging stage. FIG. 4 shows in the top graph at100 minutes the constant voltage charge rate over time. The next lowergraph at 100 minutes is the constant current charge rate over time. Theequivalent charging power is indicated in the next lower graph and thebottom graph is the equivalent resistance.

For an electric vehicle (EV) charging system, the first thing for an EVwireless charger to determine is the state of charge (SOC) of thebattery so as to choose either CC mode or CV mode to charge the batteryproperly. To ensure that the battery is charged in the right mode at alltimes, it is assumed that the wireless charging system has a chargingcontroller at the receiver side which can convert the received ac powerinto dc power, and can also charge the battery according to the batterycharging profile. The controller is able to automatically adjust itsinput impedance to absorb the correct power from the receiver anddeliver the charge to the battery.

With the help of the charging controller at the receiver side, what thewireless charging system needs to do is to predict the power needed fromthe battery and change its operating conditions (e.g. adjust itsoperating frequency and input voltage/input current in the transmittercircuit so that the wireless power is transferred to the load (such asthe battery) in the most efficient and optimal manner. This is describedin the article, J. Yin et al., “A Systematic Approach for LoadMonitoring and Power Control in Wireless Power Transfer Systems WithoutAny Direct Output Measurement”, IEEE Transactions onPower Electronics,vol. 30, pp. 1657-1667, 2015 (the “Yin 2015 article”), which isincorporated herein in its entirety.

To ensure the wireless charging system always operates in the optimalmode, all of the parameters of the whole system need to be known. Themost difficult parameters are the mutual inductance(s) related to (1)coil misalignment and (2) the distances between the transmitting coil(s)and the receiver coil(s). This is particularly true for wireless EVcharging, because it is not easy to park a vehicle in the exact locationeach time. There must be some tolerance for misalignment between thetransmitter coil and the receiver coil. In addition, the distancebetween the transmitter coil and the receiver coil may also vary fromone type of vehicle to another type. A 4-wheel-drive vehicle willprobably have a large distance between the transmitter coil and thereceiver coil, while a compact car may have a smaller distance.

Since the uncertainties of the (1) coil misalignment and (2) coildistance are related to the mutual inductance values of the mutuallycoupled coils only, and are independent of the capacitance in the coilresonators, the present invention focuses on a method of estimatingthese mutual inductance terms in a rapid manner. The assumption is thatthe self-inductance terms and the resonant-capacitance terms are known(obtained either from the method of the Hui '775 PCT application or fromthe manufacturers). Once determined for a particular setup, they neednot be determined again. Only the mutual inductance terms and the loadimpedance terms are unknown and must be determined.

The determination of the mutual inductance terms is based on information(such as input voltage, input current and the phase angle between theinput voltage and current waveforms) obtained only on the transmittercircuit without using any direct measurement on the receiver circuit.This important feature eliminates the need for a wireless communicationsystem between the transmitter and receiver circuits. For theapplication of EV charging, there is no need to know the exact positionsof or relative positions between the transmitter coil and the receivercoil, because such information is represented in the electrical circuitmodel in terms of the mutual inductance.

The present invention proposes a method of identifying the systemparameters for EV wireless charging, but the principles can be appliedto any wireless power transfer system with 2 or more mutually coupledcoils. Examples are the 2-coil EV wireless charging system in FIG. 5 andthe 3-coil system shown in FIG. 6. In both drawings the ac source isshown generating the voltage Vs which is applied to input coil A, whichhas a resonance capacitance C₁. The field of coil A is coupled to outputcoil B, which has resonance capacitance C₂. The ac voltage from coil Bis converted into dc voltage in converter 50, the dc from converter 50is converted into a different dc voltage in converter 52, whose outputis applied to the load. Thus, in FIG. 5 there is mutual coupling M₁₂between coils A and B.

In the 3-coil system of FIG. 6, there is an additional relaycoil-resonator D with a resonance capacitance C₀, but it is otherwisesimilar to the system of FIG. 5. Because of the relay coil there areadditional mutual couplings M₁₀ and M₀₂.

A more generalized analysis of the system can be explained with theassistance of FIG. 7 which shows a wireless power transfer systemconsisting of n coils, where the 1^(st) coil A is the transmitter, then^(th) coil B is the receiver. The relay coils are D₂ up to D_(n−1). IfL_(i) is the self-inductance, R_(i) is the coil resistance and C_(i) isthe resonant capacitance of the i^(th) coil respectively; M_(ij) is themutual-inductance between the i^(th) coil and the j^(th) coil (obviouslyM_(ij)=M_(ji)), and Z_(L) is the load impedance, then the system couldbe described in a general matrix equation (1).

$\begin{matrix}{\begin{bmatrix}{V_{s}(\omega)} \\0 \\\vdots \\0 \\0\end{bmatrix} = {\quad{\begin{bmatrix}{Z_{1}(\omega)} & {j\; \omega \; M_{12}} & \ldots & {\; {j\; \omega \; M_{1{({n - 1})}}}} & {j\; \omega \; M_{1n}} \\{j\; \omega \; M_{12}} & {Z_{2}(\omega)} & \ldots & {j\; \omega \; M_{2{({n - 1})}}} & {j\; \omega \; M_{2n}} \\\vdots & \vdots & \ddots & \vdots & \vdots \\{j\; \omega \; M_{1{({n - 1})}}} & {j\; \omega \; M_{2{({n - 1})}}} & \ldots & {Z_{n - 1}(\omega)} & {j\; \omega \; M_{{({n - 1})}n}} \\{j\; \omega \; M_{1n}} & {j\; \omega \; M_{2n}} & \ldots & {j\; \omega \; M_{{({n - 1})}n}} & {{Z_{n}(\omega)} + {Z_{L}(\omega)}}\end{bmatrix}\begin{bmatrix}{I_{1}(\omega)} \\{I_{2}(\omega)} \\\vdots \\{I_{n - 1}(\omega)} \\{I_{n}(\omega)}\end{bmatrix}}}} & (1)\end{matrix}$

where

${Z_{i}(\omega)} = {R_{i} + {j\left( {{\omega \; L_{i}} - \frac{1}{\omega \; C_{i}}} \right)}}$

is the total impedance of the i^(th) coil, V_(S)(ω) is the input voltagevector in the first coil, I_(i)(ω) is the current vector of the i^(th)coil, and ω is the angular frequency of V_(S)(ω).

In the previous invention disclosed in the Hui '775 PCT application,which is incorporated herein in its entity, the present inventorsdemonstrated the use of a 2-stage process to (i) obtain the systemparameters including the mutual inductance terms based on informationobtained from the transmitter circuit and (ii) take a measurement of theinput voltage and input current, calculate the load impedance andcontrol the output voltage and output power in the receiver circuit.This technique makes use of a search algorithm, such as the geneticalgorithm or the particle swarm algorithm, in stage (i). However,solving those algorithms requires typically tens of minutes to obtainthe results. The advantage of this previous invention is that once theself-inductance terms and resonant-capacitance terms are determined,these values of a particular system (such as an EV) remain the same. Theonly terms that may change each time are the mutual inductance terms,which depend on the vehicle parking location with respect to thecharging coils.

The present invention maintains the same concept of using informationonly on the transmitter circuit. However, the present invention proposesthe use of a much faster method to obtain the mutual inductance terms.In general, it is necessary to have the number of equations matching orlarger than the number of unknowns in order to solve the equations forall the unknowns analytically. Sometimes there is no analytical solutionor redundant equations need to be used to reduce the effect ofmeasurement errors. Search algorithms are usually used to estimate theunknowns under such situation. With the present invention, the wirelesspower system is excited at a number of different frequencies (oralternatively different input voltage levels in some cases) in order toobtain a sufficient number of equations to meet the number of unknowns.In this way, all of the unknowns can be solved analytically ornumerically, without using any time-consuming search algorithms such asgenetic algorithms or particle swarm algorithms. Therefore, this newapproach is a much faster method than using search algorithms.

Now consider the system matrix equation (1). It is assumed that only theinput voltage V_(S)(ω) and input current I₁(ω) can be measured and usedin the parameter identification process. If it is assumed that all ofthe self-inductance terms and the capacitance terms are known (obtainedeither from the method of the inventors' previous invention or from themanufacturers), then z_(i)(ω) is known when the frequency is known.

The unknowns in equation (1) are: M_(ij) (i, j=1, 2, . . . n; i≠j),I₂(ω)˜I_(n)(ω), and Z_(L)(ω). For an EV wireless charger the load can betreated as a constant pure resistive load or a pure resistive load whichabsorbs constant power during short periods of time, thus Z_(L)(ω) couldbe R_(L). Hence, there are totally

${\begin{pmatrix}n \\2\end{pmatrix} + \left( {n - 1} \right) + 1} = \frac{n\left( {n + 1} \right)}{2}$

unknowns when one measurement is taken for V_(S)(ω) and I₁(ω), where

$\quad\begin{pmatrix}n \\2\end{pmatrix}$

is the 2-combinations among n coils.

Obviously, for the 1^(st) measurement, there are

$\frac{n\left( {n + 1} \right)}{2}$

unknowns, but only n equations in matrix equation (1). Because there aremore unknowns than equations, it is not possible to calculate all theunknowns. Nevertheless, if more measurements are taken at differentfrequencies ω_(i), for each more measurement, there are n moreequations, but (n−1) more unknowns (1₂(ω_(i))˜I_(n)(ω_(i))). Therefore,there will be enough equations to solve the mathematical problemanalytically when

$m = {{\frac{n\left( {n + 1} \right)}{2} - n + 1} = \frac{n^{2} - n + 2}{2}}$

times measurements are taken at different frequencies as shown in matrixequation (2).

$\begin{matrix}\begin{pmatrix}{\begin{bmatrix}{V_{s}\left( \omega_{1} \right)} \\0 \\\vdots \\0 \\0\end{bmatrix} = {\begin{bmatrix}{Z_{1}\left( \omega_{1} \right)} & {j\; \omega \; M_{12}} & \ldots & {j\; \omega_{1}M_{1{({n - 1})}}} & {j\; \omega_{1}M_{1n}} \\{j\; \omega_{1}M_{12}} & {Z_{2}\left( \omega_{1} \right)} & \ldots & {j\; \omega_{1}M_{2{({n - 1})}}} & {j\; \omega_{1}M_{2n}} \\\vdots & \vdots & \ddots & \vdots & \vdots \\{j\; \omega_{1}M_{1{({n - 1})}}} & {j\; \omega_{1}M_{2{({n - 1})}}} & \ldots & {Z_{n - 1}\left( \omega_{1} \right)} & {j\; \omega_{1}M_{{({n - 1})}n}} \\{j\; \omega_{1}M_{1n}} & {j\; \omega_{1}M_{2n}} & \ldots & {j\; \omega_{1}M_{{({n - 1})}n}} & {{Z_{n}\left( \omega_{1} \right)} + R_{L}}\end{bmatrix}\begin{bmatrix}{I_{1}\left( \omega_{1} \right)} \\{I_{2}\left( \omega_{1} \right)} \\\vdots \\{I_{n - 1}\left( \omega_{1} \right)} \\{I_{n}\left( \omega_{1} \right)}\end{bmatrix}}} \\{\begin{bmatrix}{V_{s}\left( \omega_{2} \right)} \\0 \\\vdots \\0 \\0\end{bmatrix} = {\begin{bmatrix}{Z_{1}\left( \omega_{2} \right)} & {j\; \omega_{2}\; M_{12}} & \ldots & {j\; \omega_{2}M_{1{({n - 1})}}} & {j\; \omega_{2}M_{1n}} \\{j\; \omega_{2}M_{12}} & {Z_{2}\left( \omega_{2} \right)} & \ldots & {j\; \omega_{2}M_{2{({n - 1})}}} & {j\; \omega_{2}M_{2n}} \\\vdots & \vdots & \ddots & \vdots & \vdots \\{j\; \omega_{2}M_{1{({n - 1})}}} & {j\; \omega_{2}M_{2{({n - 1})}}} & \ldots & {Z_{n - 1}\left( \omega_{2} \right)} & {j\; \omega_{2}M_{{({n - 1})}n}} \\{j\; \omega_{2}M_{1n}} & {j\; \omega_{2}M_{2n}} & \ldots & {j\; \omega_{2}M_{{({n - 1})}n}} & {{Z_{n}\left( \omega_{2} \right)} + R_{L}}\end{bmatrix}\begin{bmatrix}{I_{1}\left( \omega_{2} \right)} \\{I_{2}\left( \omega_{2} \right)} \\\vdots \\{I_{n - 1}\left( \omega_{2} \right)} \\{I_{n}\left( \omega_{2} \right)}\end{bmatrix}}} \\\vdots \\{\begin{bmatrix}{V_{s}\left( \omega_{m} \right)} \\0 \\\vdots \\0 \\0\end{bmatrix} = {\begin{bmatrix}{Z_{1}\left( \omega_{m} \right)} & {j\; \omega_{m}\; M_{12}} & \ldots & {j\; \omega_{m}M_{1{({n - 1})}}} & {j\; \omega_{m}M_{1n}} \\{j\; \omega_{m}M_{12}} & {Z_{2}\left( \omega_{m} \right)} & \ldots & {j\; \omega_{m}M_{2{({n - 1})}}} & {j\; \omega_{m}M_{2n}} \\\vdots & \vdots & \ddots & \vdots & \vdots \\{j\; \omega_{m}M_{1{({n - 1})}}} & {j\; \omega_{m}M_{2{({n - 1})}}} & \ldots & {Z_{n - 1}\left( \omega_{m} \right)} & {j\; \omega_{m}M_{{({n - 1})}n}} \\{j\; \omega_{m}M_{1n}} & {j\; \omega_{m}M_{2n}} & \ldots & {j\; \omega_{m}M_{{({n - 1})}n}} & {{Z_{n}\left( \omega_{m} \right)} + R_{L}}\end{bmatrix}\begin{bmatrix}{I_{1}\left( \omega_{m} \right)} \\{I_{2}\left( \omega_{m} \right)} \\\vdots \\{I_{n - 1}\left( \omega_{m} \right)} \\{I_{n}\left( \omega_{m} \right)}\end{bmatrix}}}\end{pmatrix} & (2)\end{matrix}$

When there are only 2 coils, 1 transmitter and 1 receiver, in the EVwireless charging system with one transmitter coil and one receiver coil(FIG. 5), the matrix equations (1) and (2) can be modified to matrixequation (3) and (4).

$\begin{matrix}{\begin{bmatrix}{V_{s}(\omega)} \\0\end{bmatrix} = {\begin{bmatrix}{Z_{1}(\omega)} & {j\; \omega \; M_{12}} \\{j\; \omega \; M_{12}} & {{Z_{2}(\omega)} + R_{L}}\end{bmatrix}\begin{bmatrix}{I_{1}(\omega)} \\{I_{2}(\omega)}\end{bmatrix}}} & (3) \\\begin{pmatrix}{\begin{bmatrix}{V_{s}\left( \omega_{1} \right)} \\0\end{bmatrix} = {\begin{bmatrix}{Z_{1}\left( \omega_{1} \right)} & {j\; \omega_{1}\; M_{12}} \\{j\; \omega_{1}M_{12}} & {{Z_{2}\left( \omega_{1} \right)} + R_{L}}\end{bmatrix}\begin{bmatrix}{I_{1}\left( \omega_{1} \right)} \\{I_{2}\left( \omega_{1} \right)}\end{bmatrix}}} \\{\begin{bmatrix}{V_{s}\left( \omega_{2} \right)} \\0\end{bmatrix} = {\begin{bmatrix}{Z_{1}\left( \omega_{2} \right)} & {j\; \omega_{2}\; M_{12}} \\{j\; \omega_{2}M_{12}} & {{Z_{2}\left( \omega_{2} \right)} + R_{L}}\end{bmatrix}\begin{bmatrix}{I_{1}\left( \omega_{2} \right)} \\{I_{2}\left( \omega_{2} \right)}\end{bmatrix}}}\end{pmatrix} & (4)\end{matrix}$

In equation (4), when the input voltage V_(S)(ω₁), V_(S)(ω₂) and theinput current I₁(ω₁), I₁(ω₂) are measured at two angular frequencies(i.e. ω₁ and ω₂), there are 4 unknowns: M₁₂, R_(L), I₂(ω₁) and I₂(ω₂),and the solutions for M₁₂ and R_(L) can be determined easily.

When there are 3 coils in the EV wireless charging system including atransmitter coil A, a relay resonator coil D and a receiver coil B (FIG.6), n=3,

${m = {\frac{n^{2} - n + 2}{2} = 4}},$

the matrix equations (1) and (2) can be modified to matrix equation (5)and (6).

$\begin{matrix}{\begin{bmatrix}{V_{s}(\omega)} \\0 \\0\end{bmatrix} = {\begin{bmatrix}{Z_{1}(\omega)} & {j\; \omega \; M_{10}} & {j\; \omega \; M_{12}} \\{j\; \omega \; M_{10}} & {Z_{0}(\omega)} & {j\; \omega \; M_{02}} \\{j\; \omega \; M_{12}} & {j\; \omega \; M_{02}} & {{Z_{2}(\omega)} + R_{L}}\end{bmatrix}\begin{bmatrix}{I_{1}(\omega)} \\{I_{0}(\omega)} \\{I_{2}(\omega)}\end{bmatrix}}} & (5) \\\begin{pmatrix}{\begin{bmatrix}{V_{s}\left( \omega_{1} \right)} \\0 \\0\end{bmatrix} = {\begin{bmatrix}{Z_{1}\left( \omega_{1} \right)} & {j\; \omega_{1}\; M_{10}} & {j\; \omega_{1}\; M_{12}} \\{j\; \omega_{1}M_{10}} & {Z_{0}\left( \omega_{1} \right)} & {j\; \omega_{1}M_{02}} \\{j\; \omega_{1}M_{12}} & {j\; \omega_{1}\; M_{02}} & {{Z_{2}\left( \omega_{1} \right)} + R_{L}}\end{bmatrix}\begin{bmatrix}{I_{1}\left( \omega_{1} \right)} \\{I_{0}\left( \omega_{1} \right)} \\{I_{2}\left( \omega_{1} \right)}\end{bmatrix}}} \\{\begin{bmatrix}{V_{s}\left( \omega_{2} \right)} \\0 \\0\end{bmatrix} = {\begin{bmatrix}{Z_{1}\left( \omega_{2} \right)} & {j\; \omega_{2}\; M_{10}} & {j\; \omega_{2}\; M_{12}} \\{j\; \omega_{2}M_{10}} & {Z_{0}\left( \omega_{2} \right)} & {j\; \omega_{2}M_{02}} \\{j\; \omega_{2}M_{12}} & {j\; \omega_{2}\; M_{02}} & {{Z_{2}\left( \omega_{2} \right)} + R_{L}}\end{bmatrix}\begin{bmatrix}{I_{1}\left( \omega_{2} \right)} \\{I_{0}\left( \omega_{2} \right)} \\{I_{2}\left( \omega_{2} \right)}\end{bmatrix}}} \\{\begin{bmatrix}{V_{s}\left( \omega_{3} \right)} \\0 \\0\end{bmatrix} = {\begin{bmatrix}{Z_{1}\left( \omega_{3} \right)} & {j\; \omega_{3}\; M_{10}} & {j\; \omega_{3}\; M_{12}} \\{j\; \omega_{3}M_{10}} & {Z_{0}\left( \omega_{3} \right)} & {j\; \omega_{3}M_{02}} \\{j\; \omega_{3}M_{12}} & {j\; \omega_{3}\; M_{02}} & {{Z_{2}\left( \omega_{3} \right)} + R_{L}}\end{bmatrix}\begin{bmatrix}{I_{1}\left( \omega_{3} \right)} \\{I_{0}\left( \omega_{3} \right)} \\{I_{2}\left( \omega_{3} \right)}\end{bmatrix}}} \\{\begin{bmatrix}{V_{s}\left( \omega_{4} \right)} \\0 \\0\end{bmatrix} = {\begin{bmatrix}{Z_{1}\left( \omega_{4} \right)} & {j\; \omega_{4}\; M_{10}} & {j\; \omega_{4}\; M_{12}} \\{j\; \omega_{4}M_{10}} & {Z_{0}\left( \omega_{4} \right)} & {j\; \omega_{4}M_{02}} \\{j\; \omega_{4}M_{12}} & {j\; \omega_{4}\; M_{02}} & {{Z_{2}\left( \omega_{4} \right)} + R_{L}}\end{bmatrix}\begin{bmatrix}{I_{1}\left( \omega_{4} \right)} \\{I_{0}\left( \omega_{4} \right)} \\{I_{2}\left( \omega_{4} \right)}\end{bmatrix}}}\end{pmatrix} & (6)\end{matrix}$

In equation (6), when the input voltage V_(S)(ω₁) through V_(S)(ω₄) andinput current I₁(ω₁) through I₁(ω₄) are measured at 4 different angularfrequencies (i.e. ω₁ to ω₄), the result is 12 equations as shown above.At the same time, there are 12 unknowns: M₁₀, M₀₂, M₁₂, R_(L), I₀(ω₁) toI₀(ω₄), I₂(ω₁) to I₂(ω₄), so the solutions for M₁₀, M₀₂, M₁₂ and R_(L)are obtained analytically.

When the ac/dc and dc/dc converters 50, 52 are included in the receivercircuit (e.g. for a wireless charging system) as shown in FIG. 5 andFIG. 6, a simple way to describe the EV charger system's behavior is totreat the battery as a constant power load during short periods of time.In that case the constant resistive load R_(L) must be changed into aconstant power load P_(L) in equation (4) and equation (6), where

${R_{L} = \frac{P_{L}}{{I_{L}}^{2}}},{I_{L}}$

is the absolute value of load current (|I_(L)|=|I₂|). The result isequation (7) and equation (8).

$\begin{matrix}\begin{pmatrix}{\begin{bmatrix}{V_{s}\left( \omega_{1} \right)} \\0\end{bmatrix} = {\begin{bmatrix}{Z_{1}\left( \omega_{1} \right)} & {j\; \omega_{1}\; M_{12}} \\{j\; \omega_{1}M_{12}} & {{Z_{2}\left( \omega_{1} \right)} + \frac{P_{L}}{{{I_{2}\left( \omega_{1} \right)}}^{2}}}\end{bmatrix}\begin{bmatrix}{I_{1}\left( \omega_{1} \right)} \\{I_{2}\left( \omega_{1} \right)}\end{bmatrix}}} \\{\begin{bmatrix}{V_{s}\left( \omega_{2} \right)} \\0\end{bmatrix} = {\begin{bmatrix}{Z_{1}\left( \omega_{2} \right)} & {j\; \omega_{2}\; M_{12}} \\{j\; \omega_{2}M_{12}} & {{Z_{2}\left( \omega_{2} \right)} + \frac{P_{L}}{{{I_{2}\left( \omega_{2} \right)}}^{2}}}\end{bmatrix}\begin{bmatrix}{I_{1}\left( \omega_{2} \right)} \\{I_{2}\left( \omega_{2} \right)}\end{bmatrix}}}\end{pmatrix} & (7) \\\begin{pmatrix}{\begin{bmatrix}{V_{s}\left( \omega_{1} \right)} \\0 \\0\end{bmatrix} = {\begin{bmatrix}{Z_{1}\left( \omega_{1} \right)} & {j\; \omega_{1}M_{10}} & {j\; \omega_{1}M_{12}} \\{j\; \omega_{1}\; M_{10}} & {Z_{0}\left( \omega_{1} \right)} & {j\; \omega_{1}M_{02}} \\{j\; \omega_{1}\; M_{12}} & {j\; \omega_{1}\; M_{02}} & {{Z_{2}\left( \omega_{1} \right)} + \frac{P_{L}}{{{I_{2}\left( \omega_{1} \right)}}^{2}}}\end{bmatrix}\begin{bmatrix}{I_{1}\left( \omega_{1} \right)} \\{I_{0}\left( \omega_{1} \right)} \\{I_{2}\left( \omega_{1} \right)}\end{bmatrix}}} \\{\begin{bmatrix}{V_{s}\left( \omega_{2} \right)} \\0 \\0\end{bmatrix} = {\begin{bmatrix}{Z_{1}\left( \omega_{2} \right)} & {j\; \omega_{2}M_{10}} & {j\; \omega_{2}M_{12}} \\{j\; \omega_{2}\; M_{10}} & {Z_{0}\left( \omega_{2} \right)} & {j\; \omega_{2}M_{02}} \\{j\; \omega_{2}\; M_{12}} & {j\; \omega_{2}\; M_{02}} & {{Z_{2}\left( \omega_{2} \right)} + \frac{P_{L}}{{{I_{2}\left( \omega_{2} \right)}}^{2}}}\end{bmatrix}\begin{bmatrix}{I_{1}\left( \omega_{2} \right)} \\{I_{0}\left( \omega_{2} \right)} \\{I_{2}\left( \omega_{2} \right)}\end{bmatrix}}} \\{\begin{bmatrix}{V_{s}\left( \omega_{3} \right)} \\0 \\0\end{bmatrix} = {\begin{bmatrix}{Z_{1}\left( \omega_{3} \right)} & {j\; \omega_{3}M_{10}} & {j\; \omega_{3}M_{12}} \\{j\; \omega_{3}\; M_{10}} & {Z_{0}\left( \omega_{3} \right)} & {j\; \omega_{3}M_{02}} \\{j\; \omega_{3}\; M_{12}} & {j\; \omega_{3}\; M_{02}} & {{Z_{2}\left( \omega_{3} \right)} + \frac{P_{L}}{{{I_{2}\left( \omega_{3} \right)}}^{2}}}\end{bmatrix}\begin{bmatrix}{I_{1}\left( \omega_{3} \right)} \\{I_{0}\left( \omega_{3} \right)} \\{I_{2}\left( \omega_{3} \right)}\end{bmatrix}}} \\{\begin{bmatrix}{V_{s}\left( \omega_{4} \right)} \\0 \\0\end{bmatrix} = {\begin{bmatrix}{Z_{1}\left( \omega_{4} \right)} & {j\; \omega_{4}M_{10}} & {j\; \omega_{4}M_{12}} \\{j\; \omega_{4}\; M_{10}} & {Z_{0}\left( \omega_{4} \right)} & {j\; \omega_{4}M_{02}} \\{j\; \omega_{4}\; M_{12}} & {j\; \omega_{4}\; M_{02}} & {{Z_{2}\left( \omega_{4} \right)} + \frac{P_{L}}{{{I_{2}\left( \omega_{4} \right)}}^{2}}}\end{bmatrix}\begin{bmatrix}{I_{1}\left( \omega_{4} \right)} \\{I_{0}\left( \omega_{4} \right)} \\{I_{2}\left( \omega_{4} \right)}\end{bmatrix}}}\end{pmatrix} & (8)\end{matrix}$

Comparing equation (7) to equation (3) and equation (8) to equation (4),it can be seen that the number of unknowns and the number of equationsare the same for each pair of equations. Therefore, theoretically,equation (7) can be solved analytically to get the parameter values forM₁₂, P_(L), I₂(ω₁) and I₂(ω₂). Similarly, equation (8) can be solved toget the parameter values for M₁₀, M₀₂, M₁₂, P_(L), I₀(ω₁) to I₀(ω₄), andI₂(ω₁) to I₂(ω₄).

Please note that, in equation (7) and equation (8), all of the unknowncurrents as well as the measured input voltages and currents are complexvalues. Thus, it is not a straightforward process to get the solutionsto these complex matrix equations when the absolute values like|I_(i)(ω_(i))| are involved. To simplify the process of calculation, wecan split or decouple each complex equation into its real part and itsimaginary part to form two ordinary equations. Then equation (7) andequation (8) can be rewritten as equation (9) and equation (10),respectively.

$\begin{matrix}\begin{pmatrix}{\begin{bmatrix}{{Re}\left( {V_{s}\left( \omega_{1} \right)} \right)} \\0\end{bmatrix} = {{\begin{bmatrix}{{Re}\left( {Z_{1}\left( \omega_{1} \right)} \right)} & 0 \\0 & {{{Re}\left( {Z_{2}\left( \omega_{1} \right)} \right)} + \frac{P_{L}}{{{Re}\left( {I_{2}\left( \omega_{1} \right)} \right)}^{2} + {{Im}\left( {I_{2}\left( \omega_{1} \right)} \right)}^{2}}}\end{bmatrix}\begin{bmatrix}{{Re}\left( {I_{1}\left( \omega_{1} \right)} \right)} \\{{Re}\left( {I_{2}\left( \omega_{1} \right)} \right)}\end{bmatrix}} - {\begin{bmatrix}{{Im}\left( {Z_{1}\left( \omega_{1} \right)} \right)} & {\omega_{1}M_{12}} \\{\omega_{1}M_{12}} & {{Im}\left( {Z_{2}\left( \omega_{1} \right)} \right)}\end{bmatrix}\begin{bmatrix}{{Im}\left( {I_{1}\left( \omega_{1} \right)} \right)} \\{{Im}\left( {I_{2}\left( \omega_{1} \right)} \right)}\end{bmatrix}}}} \\{\begin{bmatrix}{{Im}\left( {V_{s}\left( \omega_{1} \right)} \right)} \\0\end{bmatrix} = {{\begin{bmatrix}{{Re}\left( {Z_{1}\left( \omega_{1} \right)} \right)} & 0 \\0 & {{{Re}\left( {Z_{2}\left( \omega_{1} \right)} \right)} + \frac{P_{L}}{{{Re}\left( {I_{2}\left( \omega_{1} \right)} \right)}^{2} + {{Im}\left( {I_{2}\left( \omega_{1} \right)} \right)}^{2}}}\end{bmatrix}\begin{bmatrix}{{Im}\left( {I_{1}\left( \omega_{1} \right)} \right)} \\{{Im}\left( {I_{2}\left( \omega_{1} \right)} \right)}\end{bmatrix}} + {\begin{bmatrix}{{Im}\left( {Z_{1}\left( \omega_{1} \right)} \right)} & {\omega_{1}M_{12}} \\{\omega_{1}M_{12}} & {{Im}\left( {Z_{2}\left( \omega_{1} \right)} \right)}\end{bmatrix}\begin{bmatrix}{{Re}\left( {I_{1}\left( \omega_{1} \right)} \right)} \\{{Re}\left( {I_{2}\left( \omega_{1} \right)} \right)}\end{bmatrix}}}} \\{\begin{bmatrix}{{Re}\left( {V_{s}\left( \omega_{2} \right)} \right)} \\0\end{bmatrix} = {{\begin{bmatrix}{{Re}\left( {Z_{1}\left( \omega_{2} \right)} \right)} & 0 \\0 & {{{Re}\left( {Z_{2}\left( \omega_{2} \right)} \right)} + \frac{P_{L}}{{{Re}\left( {I_{2}\left( \omega_{2} \right)} \right)}^{2} + {{Im}\left( {I_{2}\left( \omega_{2} \right)} \right)}^{2}}}\end{bmatrix}\begin{bmatrix}{{Re}\left( {I_{1}\left( \omega_{2} \right)} \right)} \\{{Re}\left( {I_{2}\left( \omega_{2} \right)} \right)}\end{bmatrix}} - {\begin{bmatrix}{{Im}\left( {Z_{1}\left( \omega_{2} \right)} \right)} & {\omega_{2}M_{12}} \\{\omega_{2}M_{12}} & {{Im}\left( {Z_{2}\left( \omega_{2} \right)} \right)}\end{bmatrix}\begin{bmatrix}{{Im}\left( {I_{1}\left( \omega_{2} \right)} \right)} \\{{Im}\left( {I_{2}\left( \omega_{2} \right)} \right)}\end{bmatrix}}}} \\{\begin{bmatrix}{{Im}\left( {V_{s}\left( \omega_{2} \right)} \right)} \\0\end{bmatrix} = {{\begin{bmatrix}{{Re}\left( {Z_{1}\left( \omega_{2} \right)} \right)} & 0 \\0 & {{{Re}\left( {Z_{2}\left( \omega_{2} \right)} \right)} + \frac{P_{L}}{{{Re}\left( {I_{2}\left( \omega_{2} \right)} \right)}^{2} + {{Im}\left( {I_{2}\left( \omega_{2} \right)} \right)}^{2}}}\end{bmatrix}\begin{bmatrix}{{Im}\left( {I_{1}\left( \omega_{2} \right)} \right)} \\{{Im}\left( {I_{2}\left( \omega_{2} \right)} \right)}\end{bmatrix}} + {\begin{bmatrix}{{Im}\left( {Z_{1}\left( \omega_{2} \right)} \right)} & {\omega_{2}M_{12}} \\{\omega_{2}M_{12}} & {{Im}\left( {Z_{2}\left( \omega_{2} \right)} \right)}\end{bmatrix}\begin{bmatrix}{{Re}\left( {I_{1}\left( \omega_{2} \right)} \right)} \\{{Re}\left( {I_{2}\left( \omega_{2} \right)} \right)}\end{bmatrix}}}}\end{pmatrix} & (9) \\\begin{pmatrix}{\begin{bmatrix}{{Re}\left( {V_{s}\left( \omega_{1} \right)} \right)} \\0 \\0\end{bmatrix} = {{\begin{bmatrix}{{Re}\left( {Z_{1}\left( \omega_{1} \right)} \right)} & 0 & 0 \\0 & {{Re}\left( {Z_{0}\left( \omega_{1} \right)} \right)} & 0 \\0 & 0 & \begin{matrix}{{{Re}\left( {Z_{2}\left( \omega_{1} \right)} \right)} +} \\\frac{P_{L}}{{{Re}\left( {I_{2}\left( \omega_{1} \right)} \right)}^{2} + {{Im}\left( {I_{2}\left( \omega_{1} \right)} \right)}^{2}}\end{matrix}\end{bmatrix}\begin{bmatrix}{{Re}\left( {I_{1}\left( \omega_{1} \right)} \right)} \\{{Re}\left( {I_{0}\left( \omega_{1} \right)} \right)} \\{{Re}\left( {I_{2}\left( \omega_{1} \right)} \right)}\end{bmatrix}} - {\begin{bmatrix}{{Im}\left( {Z_{1}\left( \omega_{1} \right)} \right)} & {\omega_{1}M_{10}} & {\omega_{1}M_{12}} \\{\omega_{1}M_{10}} & {{Re}\left( {Z_{0}\left( \omega_{1} \right)} \right)} & {\omega_{1}M_{02}} \\{\omega_{1}M_{12}} & {\omega_{1}M_{02}} & {{Im}\left( {Z_{2}\left( \omega_{1} \right)} \right)}\end{bmatrix}\begin{bmatrix}{{Im}\left( {I_{1}\left( \omega_{1} \right)} \right)} \\{{Im}\left( {I_{0}\left( \omega_{1} \right)} \right)} \\{{Im}\left( {I_{2}\left( \omega_{1} \right)} \right)}\end{bmatrix}}}} \\{\begin{bmatrix}{{Im}\left( {V_{s}\left( \omega_{1} \right)} \right)} \\0 \\0\end{bmatrix} = {{\begin{bmatrix}{{Re}\left( {Z_{1}\left( \omega_{1} \right)} \right)} & 0 & 0 \\0 & {{Re}\left( {Z_{0}\left( \omega_{1} \right)} \right)} & 0 \\0 & 0 & \begin{matrix}{{{Re}\left( {Z_{2}\left( \omega_{1} \right)} \right)} +} \\\frac{P_{L}}{{{Re}\left( {I_{2}\left( \omega_{1} \right)} \right)}^{2} + {{Im}\left( {I_{2}\left( \omega_{1} \right)} \right)}^{2}}\end{matrix}\end{bmatrix}\begin{bmatrix}{{Im}\left( {I_{1}\left( \omega_{1} \right)} \right)} \\{{Im}\left( {I_{0}\left( \omega_{1} \right)} \right)} \\{{Im}\left( {I_{2}\left( \omega_{1} \right)} \right)}\end{bmatrix}} + {\begin{bmatrix}{{Im}\left( {Z_{1}\left( \omega_{1} \right)} \right)} & {\omega_{1}M_{10}} & {\omega_{1}M_{12}} \\{\omega_{1}M_{10}} & {{Re}\left( {Z_{0}\left( \omega_{1} \right)} \right)} & {\omega_{1}M_{02}} \\{\omega_{1}M_{12}} & {\omega_{1}M_{02}} & {{Im}\left( {Z_{2}\left( \omega_{1} \right)} \right)}\end{bmatrix}\begin{bmatrix}{{Re}\left( {I_{1}\left( \omega_{1} \right)} \right)} \\{{Re}\left( {I_{0}\left( \omega_{1} \right)} \right)} \\{{Re}\left( {I_{2}\left( \omega_{1} \right)} \right)}\end{bmatrix}}}} \\{\begin{bmatrix}{{Re}\left( {V_{s}\left( \omega_{2} \right)} \right)} \\0 \\0\end{bmatrix} = {{\begin{bmatrix}{{Re}\left( {Z_{1}\left( \omega_{2} \right)} \right)} & 0 & 0 \\0 & {{Re}\left( {Z_{0}\left( \omega_{2} \right)} \right)} & 0 \\0 & 0 & \begin{matrix}{{{Re}\left( {Z_{2}\left( \omega_{2} \right)} \right)} +} \\\frac{P_{L}}{{{Re}\left( {I_{2}\left( \omega_{2} \right)} \right)}^{2} + {{Im}\left( {I_{2}\left( \omega_{2} \right)} \right)}^{2}}\end{matrix}\end{bmatrix}\begin{bmatrix}{{Re}\left( {I_{1}\left( \omega_{2} \right)} \right)} \\{{Re}\left( {I_{0}\left( \omega_{2} \right)} \right)} \\{{Re}\left( {I_{2}\left( \omega_{2} \right)} \right)}\end{bmatrix}} - {\begin{bmatrix}{{Im}\left( {Z_{1}\left( \omega_{2} \right)} \right)} & {\omega_{2}M_{10}} & {\omega_{2}M_{12}} \\{\omega_{2}M_{10}} & {{Re}\left( {Z_{0}\left( \omega_{2} \right)} \right)} & {\omega_{2}M_{02}} \\{\omega_{2}M_{12}} & {\omega_{2}M_{02}} & {{Im}\left( {Z_{2}\left( \omega_{2} \right)} \right)}\end{bmatrix}\begin{bmatrix}{{Im}\left( {I_{1}\left( \omega_{2} \right)} \right)} \\{{Im}\left( {I_{0}\left( \omega_{2} \right)} \right)} \\{{Im}\left( {I_{2}\left( \omega_{2} \right)} \right)}\end{bmatrix}}}} \\{\begin{bmatrix}{{Im}\left( {V_{s}\left( \omega_{2} \right)} \right)} \\0 \\0\end{bmatrix} = {{\begin{bmatrix}{{Re}\left( {Z_{1}\left( \omega_{2} \right)} \right)} & 0 & 0 \\0 & {{Re}\left( {Z_{0}\left( \omega_{2} \right)} \right)} & 0 \\0 & 0 & \begin{matrix}{{{Re}\left( {Z_{2}\left( \omega_{2} \right)} \right)} +} \\\frac{P_{L}}{{{Re}\left( {I_{2}\left( \omega_{2} \right)} \right)}^{2} + {{Im}\left( {I_{2}\left( \omega_{2} \right)} \right)}^{2}}\end{matrix}\end{bmatrix}\begin{bmatrix}{{Im}\left( {I_{1}\left( \omega_{2} \right)} \right)} \\{{Im}\left( {I_{0}\left( \omega_{2} \right)} \right)} \\{{Im}\left( {I_{2}\left( \omega_{2} \right)} \right)}\end{bmatrix}} + {\begin{bmatrix}{{Im}\left( {Z_{1}\left( \omega_{2} \right)} \right)} & {\omega_{2}M_{10}} & {\omega_{2}M_{12}} \\{\omega_{2}M_{10}} & {{Re}\left( {Z_{0}\left( \omega_{2} \right)} \right)} & {\omega_{2}M_{02}} \\{\omega_{2}M_{12}} & {\omega_{2}M_{02}} & {{Im}\left( {Z_{2}\left( \omega_{2} \right)} \right)}\end{bmatrix}\begin{bmatrix}{{Re}\left( {I_{1}\left( \omega_{2} \right)} \right)} \\{{Re}\left( {I_{0}\left( \omega_{2} \right)} \right)} \\{{Re}\left( {I_{2}\left( \omega_{2} \right)} \right)}\end{bmatrix}}}} \\{\begin{bmatrix}{{Re}\left( {V_{s}\left( \omega_{3} \right)} \right)} \\0 \\0\end{bmatrix} = {{\begin{bmatrix}{{Re}\left( {Z_{1}\left( \omega_{3} \right)} \right)} & 0 & 0 \\0 & {{Re}\left( {Z_{0}\left( \omega_{3} \right)} \right)} & 0 \\0 & 0 & \begin{matrix}{{{Re}\left( {Z_{2}\left( \omega_{3} \right)} \right)} +} \\\frac{P_{L}}{{{Re}\left( {I_{2}\left( \omega_{3} \right)} \right)}^{2} + {{Im}\left( {I_{2}\left( \omega_{3} \right)} \right)}^{2}}\end{matrix}\end{bmatrix}\begin{bmatrix}{{Re}\left( {I_{1}\left( \omega_{3} \right)} \right)} \\{{Re}\left( {I_{0}\left( \omega_{3} \right)} \right)} \\{{Re}\left( {I_{2}\left( \omega_{3} \right)} \right)}\end{bmatrix}} - {\begin{bmatrix}{{Im}\left( {Z_{1}\left( \omega_{3} \right)} \right)} & {\omega_{3}M_{10}} & {\omega_{3}M_{12}} \\{\omega_{3}M_{10}} & {{Re}\left( {Z_{0}\left( \omega_{3} \right)} \right)} & {\omega_{3}M_{02}} \\{\omega_{3}M_{12}} & {\omega_{3}M_{02}} & {{Im}\left( {Z_{2}\left( \omega_{3} \right)} \right)}\end{bmatrix}\begin{bmatrix}{{Im}\left( {I_{1}\left( \omega_{3} \right)} \right)} \\{{Im}\left( {I_{0}\left( \omega_{3} \right)} \right)} \\{{Im}\left( {I_{2}\left( \omega_{3} \right)} \right)}\end{bmatrix}}}} \\{\begin{bmatrix}{{Im}\left( {V_{s}\left( \omega_{3} \right)} \right)} \\0 \\0\end{bmatrix} = {{\begin{bmatrix}{{Re}\left( {Z_{1}\left( \omega_{3} \right)} \right)} & 0 & 0 \\0 & {{Re}\left( {Z_{0}\left( \omega_{3} \right)} \right)} & 0 \\0 & 0 & \begin{matrix}{{{Re}\left( {Z_{2}\left( \omega_{3} \right)} \right)} +} \\\frac{P_{L}}{{{Re}\left( {I_{2}\left( \omega_{3} \right)} \right)}^{2} + {{Im}\left( {I_{2}\left( \omega_{3} \right)} \right)}^{2}}\end{matrix}\end{bmatrix}\begin{bmatrix}{{Im}\left( {I_{1}\left( \omega_{3} \right)} \right)} \\{{Im}\left( {I_{0}\left( \omega_{3} \right)} \right)} \\{{Im}\left( {I_{2}\left( \omega_{3} \right)} \right)}\end{bmatrix}} + {\begin{bmatrix}{{Im}\left( {Z_{1}\left( \omega_{3} \right)} \right)} & {\omega_{3}M_{10}} & {\omega_{3}M_{12}} \\{\omega_{3}M_{10}} & {{Re}\left( {Z_{0}\left( \omega_{3} \right)} \right)} & {\omega_{3}M_{02}} \\{\omega_{3}M_{12}} & {\omega_{3}M_{02}} & {{Im}\left( {Z_{2}\left( \omega_{3} \right)} \right)}\end{bmatrix}\begin{bmatrix}{{Re}\left( {I_{1}\left( \omega_{3} \right)} \right)} \\{{Re}\left( {I_{0}\left( \omega_{3} \right)} \right)} \\{{Re}\left( {I_{2}\left( \omega_{3} \right)} \right)}\end{bmatrix}}}} \\{\begin{bmatrix}{{Re}\left( {V_{s}\left( \omega_{4} \right)} \right)} \\0 \\0\end{bmatrix} = {{\begin{bmatrix}{{Re}\left( {Z_{1}\left( \omega_{4} \right)} \right)} & 0 & 0 \\0 & {{Re}\left( {Z_{0}\left( \omega_{4} \right)} \right)} & 0 \\0 & 0 & \begin{matrix}{{{Re}\left( {Z_{2}\left( \omega_{4} \right)} \right)} +} \\\frac{P_{L}}{{{Re}\left( {I_{2}\left( \omega_{4} \right)} \right)}^{2} + {{Im}\left( {I_{2}\left( \omega_{4} \right)} \right)}^{2}}\end{matrix}\end{bmatrix}\begin{bmatrix}{{Re}\left( {I_{1}\left( \omega_{4} \right)} \right)} \\{{Re}\left( {I_{0}\left( \omega_{4} \right)} \right)} \\{{Re}\left( {I_{2}\left( \omega_{4} \right)} \right)}\end{bmatrix}} - {\begin{bmatrix}{{Im}\left( {Z_{1}\left( \omega_{4} \right)} \right)} & {\omega_{4}M_{10}} & {\omega_{4}M_{12}} \\{\omega_{4}M_{10}} & {{Re}\left( {Z_{0}\left( \omega_{4} \right)} \right)} & {\omega_{4}M_{02}} \\{\omega_{4}M_{12}} & {\omega_{4}M_{02}} & {{Im}\left( {Z_{2}\left( \omega_{4} \right)} \right)}\end{bmatrix}\begin{bmatrix}{{Im}\left( {I_{1}\left( \omega_{4} \right)} \right)} \\{{Im}\left( {I_{0}\left( \omega_{4} \right)} \right)} \\{{Im}\left( {I_{2}\left( \omega_{4} \right)} \right)}\end{bmatrix}}}} \\{\begin{bmatrix}{{Im}\left( {V_{s}\left( \omega_{4} \right)} \right)} \\0 \\0\end{bmatrix} = {{\begin{bmatrix}{{Re}\left( {Z_{1}\left( \omega_{4} \right)} \right)} & 0 & 0 \\0 & {{Re}\left( {Z_{0}\left( \omega_{4} \right)} \right)} & 0 \\0 & 0 & \begin{matrix}{{{Re}\left( {Z_{2}\left( \omega_{4} \right)} \right)} +} \\\frac{P_{L}}{{{Re}\left( {I_{2}\left( \omega_{4} \right)} \right)}^{2} + {{Im}\left( {I_{2}\left( \omega_{4} \right)} \right)}^{2}}\end{matrix}\end{bmatrix}\begin{bmatrix}{{Im}\left( {I_{1}\left( \omega_{4} \right)} \right)} \\{{Im}\left( {I_{0}\left( \omega_{4} \right)} \right)} \\{{Im}\left( {I_{2}\left( \omega_{4} \right)} \right)}\end{bmatrix}} + {\begin{bmatrix}{{Im}\left( {Z_{1}\left( \omega_{4} \right)} \right)} & {\omega_{4}M_{10}} & {\omega_{4}M_{12}} \\{\omega_{4}M_{10}} & {{Re}\left( {Z_{0}\left( \omega_{4} \right)} \right)} & {\omega_{4}M_{02}} \\{\omega_{4}M_{12}} & {\omega_{4}M_{02}} & {{Im}\left( {Z_{2}\left( \omega_{4} \right)} \right)}\end{bmatrix}\begin{bmatrix}{{Re}\left( {I_{1}\left( \omega_{4} \right)} \right)} \\{{Re}\left( {I_{0}\left( \omega_{4} \right)} \right)} \\{{Re}\left( {I_{2}\left( \omega_{4} \right)} \right)}\end{bmatrix}}}}\end{pmatrix} & (10)\end{matrix}$

In equation (9), there are 8 equations but only 6 unknowns: M₁₂, P_(L),Rc(I₂(ω₁)), Im(I₂(ω₁)) , Re(I₂(ω₂)) and Im(I₂(ω₂)), so it is sufficientto solve the equation and get the parameter values: M₁₂ and P_(L).Similarly, in equation (10), there are 24 equations and 20 unknowns:M₁₀, M₁₂, M₀₂, P_(L), Re(I₀(ω₁)), Im(I₀(ω₁)), Re(I₀(ω₂)), Im(I₀(ω₂)),Re(I₀(ω₃)), Im(I₀(ω₃)), Re(I₀(ω₄)), Im(I₀(ω₄)), Re(I₂(ω₁)), Im(I₂(ω₁)),Re(I₂(ω₂)), Im(I₂(ω₂)), Re(I₂(ω₃)), IM(I₂(ω₃)), Re(I₂(ω₄)), Im(I₂(ω₄)).Thus, it is also sufficient to calculate the parameter values within 4sets of measurements of the input voltage and input current at 4different frequencies in order to optimize the operation of the EVwireless charger

In equation (8) and equation (10), we treat M₁₀ as unknown. In practice,the second transmitter coil is the relay resonator that is used toenhance the magnetic flux. See, the Zhong 2015 article 1. As a result,the relative position and the mutual inductance of the Tx coil-1 A andthe relay coil-0 D of the 3-coil system in FIG. 3 and FIG. 6 are known.Therefore, the number of unknowns can be reduced by 1 and only 3different sets of measurements (e.g. using 3 different frequencies or 3different magnitudes of input voltage) of the input voltage and inputcurrent are necessary to identify the needed parameters of the system asshown in equation (11) and equation (12).

In equation (11), there are 9 unknowns: M₀₂, M₁₂, R_(L), I₀(ω₁) toI₀(ω₃), I₂(ω₁) to I₂(ω₃), and a total of 9 equations, so the solutionsfor M₀₂, M₁₂ and R_(L) are theoretically obtained. In equation (12),there are 18 equations but 15 unknowns: M₀₂, M₁₂, P_(L), Re(I₀(ω₁)) toRe(I₀(ω₃)), Im(I₀(ω₁)) to Im(I₀(ω₃)), Re(I₂(ω₁) to Re(I₂(0) , Im(I₂(ω₁))to Im(I₂(ω₃)). Therefore, it is sufficient to get the needed parameterswith 3 different sets of measurements of the input voltage and inputcurrent. With the parameters known, the previous method for optimizingthe operation of the wireless EV charger reported in the Hui ‘775 PCTapplication can be used for optimal control of the wireless power systemwithout using any direct measurements of the output load.

$\begin{matrix}\begin{pmatrix}{\begin{bmatrix}{V_{s}\left( \omega_{1} \right)} \\0 \\0\end{bmatrix} = {\begin{bmatrix}{Z_{1}\left( \omega_{1} \right)} & {j\; \omega_{1}\; M_{10}} & {j\; \omega_{1}\; M_{12}} \\{j\; \omega_{1}\; M_{10}} & {Z_{0}\left( \omega_{1} \right)} & {j\; \omega_{1}\; M_{02}} \\{j\; \omega_{1}M_{12}} & {j\; \omega_{1}\; M_{02}} & {{Z_{2}\left( \omega_{1} \right)} + \frac{P_{L}}{{{I_{2}\left( \omega_{1} \right)}}^{2}}}\end{bmatrix}\begin{bmatrix}{I_{1}\left( \omega_{1} \right)} \\{I_{0}\left( \omega_{1} \right)} \\{I_{2}\left( \omega_{1} \right)}\end{bmatrix}}} \\{\begin{bmatrix}{V_{s}\left( \omega_{2} \right)} \\0 \\0\end{bmatrix} = {\begin{bmatrix}{Z_{1}\left( \omega_{2} \right)} & {j\; \omega_{2}\; M_{10}} & {j\; \omega_{2}\; M_{12}} \\{j\; \omega_{2}\; M_{10}} & {Z_{0}\left( \omega_{2} \right)} & {j\; \omega_{2}\; M_{02}} \\{j\; \omega_{2}M_{12}} & {j\; \omega_{2}\; M_{02}} & {{Z_{2}\left( \omega_{2} \right)} + \frac{P_{L}}{{{I_{2}\left( \omega_{2} \right)}}^{2}}}\end{bmatrix}\begin{bmatrix}{I_{1}\left( \omega_{2} \right)} \\{I_{0}\left( \omega_{2} \right)} \\{I_{2}\left( \omega_{2} \right)}\end{bmatrix}}} \\{\begin{bmatrix}{V_{s}\left( \omega_{3} \right)} \\0 \\0\end{bmatrix} = {\begin{bmatrix}{Z_{1}\left( \omega_{3} \right)} & {j\; \omega_{3}\; M_{10}} & {j\; \omega_{3}\; M_{12}} \\{j\; \omega_{3}\; M_{10}} & {Z_{0}\left( \omega_{3} \right)} & {j\; \omega_{3}\; M_{02}} \\{j\; \omega_{3}M_{12}} & {j\; \omega_{3}\; M_{02}} & {{Z_{2}\left( \omega_{3} \right)} + \frac{P_{L}}{{{I_{2}\left( \omega_{3} \right)}}^{2}}}\end{bmatrix}\begin{bmatrix}{I_{1}\left( \omega_{3} \right)} \\{I_{0}\left( \omega_{3} \right)} \\{I_{2}\left( \omega_{3} \right)}\end{bmatrix}}}\end{pmatrix} & (11) \\\begin{pmatrix}{\begin{bmatrix}{{Re}\left( {V_{s}\left( \omega_{1} \right)} \right)} \\0 \\0\end{bmatrix} = {{\begin{bmatrix}{{Re}\left( {Z_{1}\left( \omega_{1} \right)} \right)} & 0 & 0 \\0 & {{Re}\left( {Z_{0}\left( \omega_{1} \right)} \right)} & 0 \\0 & 0 & \begin{matrix}{{{Re}\left( {Z_{2}\left( \omega_{1} \right)} \right)} +} \\\frac{P_{L}}{{{Re}\left( {I_{2}\left( \omega_{1} \right)} \right)}^{2} + {{Im}\left( {I_{2}\left( \omega_{1} \right)} \right)}^{2}}\end{matrix}\end{bmatrix}\begin{bmatrix}{{Re}\left( {I_{1}\left( \omega_{1} \right)} \right)} \\{{Re}\left( {I_{0}\left( \omega_{1} \right)} \right)} \\{{Re}\left( {I_{2}\left( \omega_{1} \right)} \right)}\end{bmatrix}} - {\begin{bmatrix}{{Im}\left( {Z_{1}\left( \omega_{1} \right)} \right)} & {\omega_{1}M_{10}} & {\omega_{1}M_{12}} \\{\omega_{1}M_{10}} & {{Re}\left( {Z_{0}\left( \omega_{1} \right)} \right)} & {\omega_{1}M_{02}} \\{\omega_{1}M_{12}} & {\omega_{1}M_{02}} & {{Im}\left( {Z_{2}\left( \omega_{1} \right)} \right)}\end{bmatrix}\begin{bmatrix}{{Im}\left( {I_{1}\left( \omega_{1} \right)} \right)} \\{{Im}\left( {I_{0}\left( \omega_{1} \right)} \right)} \\{{Im}\left( {I_{2}\left( \omega_{1} \right)} \right)}\end{bmatrix}}}} \\{\begin{bmatrix}{{Im}\left( {V_{s}\left( \omega_{1} \right)} \right)} \\0 \\0\end{bmatrix} = {{\begin{bmatrix}{{Re}\left( {Z_{1}\left( \omega_{1} \right)} \right)} & 0 & 0 \\0 & {{Re}\left( {Z_{0}\left( \omega_{1} \right)} \right)} & 0 \\0 & 0 & \begin{matrix}{{{Re}\left( {Z_{2}\left( \omega_{1} \right)} \right)} +} \\\frac{P_{L}}{{{Re}\left( {I_{2}\left( \omega_{1} \right)} \right)}^{2} + {{Im}\left( {I_{2}\left( \omega_{1} \right)} \right)}^{2}}\end{matrix}\end{bmatrix}\begin{bmatrix}{{Im}\left( {I_{1}\left( \omega_{1} \right)} \right)} \\{{Im}\left( {I_{0}\left( \omega_{1} \right)} \right)} \\{{Im}\left( {I_{2}\left( \omega_{1} \right)} \right)}\end{bmatrix}} + {\begin{bmatrix}{{Im}\left( {Z_{1}\left( \omega_{1} \right)} \right)} & {\omega_{1}M_{10}} & {\omega_{1}M_{12}} \\{\omega_{1}M_{10}} & {{Re}\left( {Z_{0}\left( \omega_{1} \right)} \right)} & {\omega_{1}M_{02}} \\{\omega_{1}M_{12}} & {\omega_{1}M_{02}} & {{Im}\left( {Z_{2}\left( \omega_{1} \right)} \right)}\end{bmatrix}\begin{bmatrix}{{Re}\left( {I_{1}\left( \omega_{1} \right)} \right)} \\{{Re}\left( {I_{0}\left( \omega_{1} \right)} \right)} \\{{Re}\left( {I_{2}\left( \omega_{1} \right)} \right)}\end{bmatrix}}}} \\{\begin{bmatrix}{{Re}\left( {V_{s}\left( \omega_{2} \right)} \right)} \\0 \\0\end{bmatrix} = {{\begin{bmatrix}{{Re}\left( {Z_{1}\left( \omega_{2} \right)} \right)} & 0 & 0 \\0 & {{Re}\left( {Z_{0}\left( \omega_{2} \right)} \right)} & 0 \\0 & 0 & \begin{matrix}{{{Re}\left( {Z_{2}\left( \omega_{2} \right)} \right)} +} \\\frac{P_{L}}{{{Re}\left( {I_{2}\left( \omega_{2} \right)} \right)}^{2} + {{Im}\left( {I_{2}\left( \omega_{2} \right)} \right)}^{2}}\end{matrix}\end{bmatrix}\begin{bmatrix}{{Re}\left( {I_{1}\left( \omega_{2} \right)} \right)} \\{{Re}\left( {I_{0}\left( \omega_{2} \right)} \right)} \\{{Re}\left( {I_{2}\left( \omega_{2} \right)} \right)}\end{bmatrix}} - {\begin{bmatrix}{{Im}\left( {Z_{1}\left( \omega_{2} \right)} \right)} & {\omega_{2}M_{10}} & {\omega_{2}M_{12}} \\{\omega_{2}M_{10}} & {{Re}\left( {Z_{0}\left( \omega_{2} \right)} \right)} & {\omega_{2}M_{02}} \\{\omega_{2}M_{12}} & {\omega_{2}M_{02}} & {{Im}\left( {Z_{2}\left( \omega_{2} \right)} \right)}\end{bmatrix}\begin{bmatrix}{{Im}\left( {I_{1}\left( \omega_{2} \right)} \right)} \\{{Im}\left( {I_{0}\left( \omega_{2} \right)} \right)} \\{{Im}\left( {I_{2}\left( \omega_{2} \right)} \right)}\end{bmatrix}}}} \\{\begin{bmatrix}{{Im}\left( {V_{s}\left( \omega_{2} \right)} \right)} \\0 \\0\end{bmatrix} = {{\begin{bmatrix}{{Re}\left( {Z_{1}\left( \omega_{2} \right)} \right)} & 0 & 0 \\0 & {{Re}\left( {Z_{0}\left( \omega_{2} \right)} \right)} & 0 \\0 & 0 & \begin{matrix}{{{Re}\left( {Z_{2}\left( \omega_{2} \right)} \right)} +} \\\frac{P_{L}}{{{Re}\left( {I_{2}\left( \omega_{2} \right)} \right)}^{2} + {{Im}\left( {I_{2}\left( \omega_{2} \right)} \right)}^{2}}\end{matrix}\end{bmatrix}\begin{bmatrix}{{Im}\left( {I_{1}\left( \omega_{2} \right)} \right)} \\{{Im}\left( {I_{0}\left( \omega_{2} \right)} \right)} \\{{Im}\left( {I_{2}\left( \omega_{2} \right)} \right)}\end{bmatrix}} + {\begin{bmatrix}{{Im}\left( {Z_{1}\left( \omega_{2} \right)} \right)} & {\omega_{2}M_{10}} & {\omega_{2}M_{12}} \\{\omega_{2}M_{10}} & {{Re}\left( {Z_{0}\left( \omega_{2} \right)} \right)} & {\omega_{2}M_{02}} \\{\omega_{2}M_{12}} & {\omega_{2}M_{02}} & {{Im}\left( {Z_{2}\left( \omega_{2} \right)} \right)}\end{bmatrix}\begin{bmatrix}{{Re}\left( {I_{1}\left( \omega_{2} \right)} \right)} \\{{Re}\left( {I_{0}\left( \omega_{2} \right)} \right)} \\{{Re}\left( {I_{2}\left( \omega_{2} \right)} \right)}\end{bmatrix}}}} \\{\begin{bmatrix}{{Re}\left( {V_{s}\left( \omega_{3} \right)} \right)} \\0 \\0\end{bmatrix} = {{\begin{bmatrix}{{Re}\left( {Z_{1}\left( \omega_{3} \right)} \right)} & 0 & 0 \\0 & {{Re}\left( {Z_{0}\left( \omega_{3} \right)} \right)} & 0 \\0 & 0 & \begin{matrix}{{{Re}\left( {Z_{2}\left( \omega_{3} \right)} \right)} +} \\\frac{P_{L}}{{{Re}\left( {I_{2}\left( \omega_{3} \right)} \right)}^{2} + {{Im}\left( {I_{2}\left( \omega_{3} \right)} \right)}^{2}}\end{matrix}\end{bmatrix}\begin{bmatrix}{{Re}\left( {I_{1}\left( \omega_{3} \right)} \right)} \\{{Re}\left( {I_{0}\left( \omega_{3} \right)} \right)} \\{{Re}\left( {I_{2}\left( \omega_{3} \right)} \right)}\end{bmatrix}} - {\begin{bmatrix}{{Im}\left( {Z_{1}\left( \omega_{3} \right)} \right)} & {\omega_{3}M_{10}} & {\omega_{3}M_{12}} \\{\omega_{3}M_{10}} & {{Re}\left( {Z_{0}\left( \omega_{3} \right)} \right)} & {\omega_{3}M_{02}} \\{\omega_{3}M_{12}} & {\omega_{3}M_{02}} & {{Im}\left( {Z_{2}\left( \omega_{3} \right)} \right)}\end{bmatrix}\begin{bmatrix}{{Im}\left( {I_{1}\left( \omega_{3} \right)} \right)} \\{{Im}\left( {I_{0}\left( \omega_{3} \right)} \right)} \\{{Im}\left( {I_{2}\left( \omega_{3} \right)} \right)}\end{bmatrix}}}} \\{\begin{bmatrix}{{Im}\left( {V_{s}\left( \omega_{3} \right)} \right)} \\0 \\0\end{bmatrix} = {{\begin{bmatrix}{{Re}\left( {Z_{1}\left( \omega_{3} \right)} \right)} & 0 & 0 \\0 & {{Re}\left( {Z_{0}\left( \omega_{3} \right)} \right)} & 0 \\0 & 0 & \begin{matrix}{{{Re}\left( {Z_{2}\left( \omega_{3} \right)} \right)} +} \\\frac{P_{L}}{{{Re}\left( {I_{2}\left( \omega_{3} \right)} \right)}^{2} + {{Im}\left( {I_{2}\left( \omega_{3} \right)} \right)}^{2}}\end{matrix}\end{bmatrix}\begin{bmatrix}{{Im}\left( {I_{1}\left( \omega_{3} \right)} \right)} \\{{Im}\left( {I_{0}\left( \omega_{3} \right)} \right)} \\{{Im}\left( {I_{2}\left( \omega_{3} \right)} \right)}\end{bmatrix}} + {\begin{bmatrix}{{Im}\left( {Z_{1}\left( \omega_{3} \right)} \right)} & {\omega_{3}M_{10}} & {\omega_{3}M_{12}} \\{\omega_{3}M_{10}} & {{Re}\left( {Z_{0}\left( \omega_{3} \right)} \right)} & {\omega_{3}M_{02}} \\{\omega_{3}M_{12}} & {\omega_{3}M_{02}} & {{Im}\left( {Z_{2}\left( \omega_{3} \right)} \right)}\end{bmatrix}\begin{bmatrix}{{Re}\left( {I_{1}\left( \omega_{3} \right)} \right)} \\{{Re}\left( {I_{0}\left( \omega_{3} \right)} \right)} \\{{Re}\left( {I_{2}\left( \omega_{3} \right)} \right)}\end{bmatrix}}}}\end{pmatrix} & (12)\end{matrix}$

Once the estimated values of the required parameters such as themutual-inductances and load resistance R_(L) or load power P_(L) havebeen obtained, there is still a possibility that the estimated values ofa certain parameter may not converge to the same value, because thecalculated values are adversely affected by system parameter toleranceand/or measurement noise. In order to obtain an estimated value withimproved accuracy, the least square approximation method can be used toobtain the best estimated values from all of estimated values derivedfrom tests at different frequencies.

In equation (7) and equation (8), different frequencies are used to getthe measurements so that there are sufficient equations to calculate theneeded system parameters. Actually, if the load is operated at constantpower mode, different input voltages can be used at the same frequency,which can provide enough power to the load and get enough equations tocalculate the system parameters as shown in equation (13) and equation(14) for 2-coil system (FIGS. 5) and 3-coil system (FIG. 6),respectively, where V_(Si)(ω₁) is the input voltage of the i^(th)measurement, I_(ji)(ω₁) is the current flow through the j^(th) coil ofthe i^(th) measurement. For equation (14), we can also use differentvoltages and/or different frequencies to form the matrix equation asshown in equation (15) to also calculate the system parameters.

$\begin{matrix}\begin{pmatrix}{\begin{bmatrix}{V_{S\; 1}\left( \omega_{1} \right)} \\0\end{bmatrix} = {\begin{bmatrix}{Z_{1}\left( \omega_{1} \right)} & {j\; \omega_{1}\; M_{12}} \\{j\; \omega_{1}M_{12}} & {{Z_{2}\left( \omega_{1} \right)} + \frac{P_{L}}{{{I_{21}\left( \omega_{1} \right)}}^{2}}}\end{bmatrix}\begin{bmatrix}{I_{11}\left( \omega_{1} \right)} \\{I_{21}\left( \omega_{1} \right)}\end{bmatrix}}} \\{\begin{bmatrix}{V_{S\; 2}\left( \omega_{1} \right)} \\0\end{bmatrix} = {\begin{bmatrix}{Z_{1}\left( \omega_{1} \right)} & {j\; \omega_{1}\; M_{12}} \\{j\; \omega_{1}M_{12}} & {{Z_{2}\left( \omega_{1} \right)} + \frac{P_{L}}{{{I_{22}\left( \omega_{1} \right)}}^{2}}}\end{bmatrix}\begin{bmatrix}{I_{12}\left( \omega_{1} \right)} \\{I_{22}\left( \omega_{1} \right)}\end{bmatrix}}}\end{pmatrix} & (13) \\\begin{pmatrix}{\begin{bmatrix}{V_{S\; 1}\left( \omega_{1} \right)} \\0 \\0\end{bmatrix} = {\begin{bmatrix}{Z_{1}\left( \omega_{1} \right)} & {j\; \omega_{1}M_{10}} & {j\; \omega_{1}M_{12}} \\{j\; \omega_{1}\; M_{10}} & {Z_{0}\left( \omega_{1} \right)} & {j\; \omega_{1}M_{02}} \\{j\; \omega_{1}\; M_{12}} & {j\; \omega_{1}\; M_{02}} & {{Z_{2}\left( \omega_{1} \right)} + \frac{P_{L}}{{{I_{21}\left( \omega_{1} \right)}}^{2}}}\end{bmatrix}\begin{bmatrix}{I_{11}\left( \omega_{1} \right)} \\{I_{01}\left( \omega_{1} \right)} \\{I_{21}\left( \omega_{1} \right)}\end{bmatrix}}} \\{\begin{bmatrix}{V_{S\; 2}\left( \omega_{1} \right)} \\0 \\0\end{bmatrix} = {\begin{bmatrix}{Z_{1}\left( \omega_{1} \right)} & {j\; \omega_{1}M_{10}} & {j\; \omega_{1}M_{12}} \\{j\; \omega_{1}\; M_{10}} & {Z_{0}\left( \omega_{1} \right)} & {j\; \omega_{1}M_{02}} \\{j\; \omega_{1}\; M_{12}} & {j\; \omega_{1}\; M_{02}} & {{Z_{2}\left( \omega_{1} \right)} + \frac{P_{L}}{{{I_{22}\left( \omega_{1} \right)}}^{2}}}\end{bmatrix}\begin{bmatrix}{I_{12}\left( \omega_{1} \right)} \\{I_{02}\left( \omega_{1} \right)} \\{I_{22}\left( \omega_{1} \right)}\end{bmatrix}}} \\{\begin{bmatrix}{V_{S\; 3}\left( \omega_{1} \right)} \\0 \\0\end{bmatrix} = {\begin{bmatrix}{Z_{1}\left( \omega_{1} \right)} & {j\; \omega_{1}M_{10}} & {j\; \omega_{1}M_{12}} \\{j\; \omega_{1}\; M_{10}} & {Z_{0}\left( \omega_{1} \right)} & {j\; \omega_{1}M_{02}} \\{j\; \omega_{1}\; M_{12}} & {j\; \omega_{1}\; M_{02}} & {{Z_{2}\left( \omega_{1} \right)} + \frac{P_{L}}{{{I_{23}\left( \omega_{1} \right)}}^{2}}}\end{bmatrix}\begin{bmatrix}{I_{13}\left( \omega_{1} \right)} \\{I_{03}\left( \omega_{1} \right)} \\{I_{23}\left( \omega_{1} \right)}\end{bmatrix}}} \\{\begin{bmatrix}{V_{S\; 4}\left( \omega_{1} \right)} \\0 \\0\end{bmatrix} = {\begin{bmatrix}{Z_{1}\left( \omega_{1} \right)} & {j\; \omega_{1}M_{10}} & {j\; \omega_{1}M_{12}} \\{j\; \omega_{1}\; M_{10}} & {Z_{0}\left( \omega_{1} \right)} & {j\; \omega_{1}M_{02}} \\{j\; \omega_{1}\; M_{12}} & {j\; \omega_{1}\; M_{02}} & {{Z_{2}\left( \omega_{1} \right)} + \frac{P_{L}}{{{I_{24}\left( \omega_{1} \right)}}^{2}}}\end{bmatrix}\begin{bmatrix}{I_{14}\left( \omega_{1} \right)} \\{I_{04}\left( \omega_{1} \right)} \\{I_{24}\left( \omega_{1} \right)}\end{bmatrix}}}\end{pmatrix} & (14) \\\begin{pmatrix}{\begin{bmatrix}{V_{S\; 1}\left( \omega_{1} \right)} \\0 \\0\end{bmatrix} = {\begin{bmatrix}{Z_{1}\left( \omega_{1} \right)} & {j\; \omega_{1}M_{10}} & {j\; \omega_{1}M_{12}} \\{j\; \omega_{1}\; M_{10}} & {Z_{0}\left( \omega_{1} \right)} & {j\; \omega_{1}M_{02}} \\{j\; \omega_{1}\; M_{12}} & {j\; \omega_{1}\; M_{02}} & {{Z_{2}\left( \omega_{1} \right)} + \frac{P_{L}}{{{I_{21}\left( \omega_{1} \right)}}^{2}}}\end{bmatrix}\begin{bmatrix}{I_{11}\left( \omega_{1} \right)} \\{I_{01}\left( \omega_{1} \right)} \\{I_{21}\left( \omega_{1} \right)}\end{bmatrix}}} \\{\begin{bmatrix}{V_{S\; 2}\left( \omega_{1} \right)} \\0 \\0\end{bmatrix} = {\begin{bmatrix}{Z_{1}\left( \omega_{1} \right)} & {j\; \omega_{1}M_{10}} & {j\; \omega_{1}M_{12}} \\{j\; \omega_{1}\; M_{10}} & {Z_{0}\left( \omega_{1} \right)} & {j\; \omega_{1}M_{02}} \\{j\; \omega_{1}\; M_{12}} & {j\; \omega_{1}\; M_{02}} & {{Z_{2}\left( \omega_{1} \right)} + \frac{P_{L}}{{{I_{22}\left( \omega_{1} \right)}}^{2}}}\end{bmatrix}\begin{bmatrix}{I_{12}\left( \omega_{1} \right)} \\{I_{02}\left( \omega_{1} \right)} \\{I_{22}\left( \omega_{1} \right)}\end{bmatrix}}} \\{\begin{bmatrix}{V_{S\; 1}\left( \omega_{2} \right)} \\0 \\0\end{bmatrix} = {\begin{bmatrix}{Z_{1}\left( \omega_{2} \right)} & {j\; \omega_{2}M_{10}} & {j\; \omega_{2}M_{12}} \\{j\; \omega_{2}\; M_{10}} & {Z_{0}\left( \omega_{2} \right)} & {j\; \omega_{2}M_{02}} \\{j\; \omega_{2}\; M_{12}} & {j\; \omega_{2}\; M_{02}} & {{Z_{2}\left( \omega_{1} \right)} + \frac{P_{L}}{{{I_{21}\left( \omega_{2} \right)}}^{2}}}\end{bmatrix}\begin{bmatrix}{I_{11}\left( \omega_{2} \right)} \\{I_{01}\left( \omega_{2} \right)} \\{I_{21}\left( \omega_{2} \right)}\end{bmatrix}}} \\{\begin{bmatrix}{V_{S\; 2}\left( \omega_{2} \right)} \\0 \\0\end{bmatrix} = {\begin{bmatrix}{Z_{1}\left( \omega_{2} \right)} & {j\; \omega_{2}M_{10}} & {j\; \omega_{2}M_{12}} \\{j\; \omega_{2}\; M_{10}} & {Z_{0}\left( \omega_{2} \right)} & {j\; \omega_{2}M_{02}} \\{j\; \omega_{2}\; M_{12}} & {j\; \omega_{2}\; M_{02}} & {{Z_{2}\left( \omega_{2} \right)} + \frac{P_{L}}{{{I_{22}\left( \omega_{2} \right)}}^{2}}}\end{bmatrix}\begin{bmatrix}{I_{12}\left( \omega_{2} \right)} \\{I_{02}\left( \omega_{2} \right)} \\{I_{22}\left( \omega_{2} \right)}\end{bmatrix}}}\end{pmatrix} & (15)\end{matrix}$

Theoretically, the solution of M₁₂ and P_(L) can be obtained fromequation (13) for the 2-coil system, and the solution of M₀₂, M₁₂ andP_(L) can be obtained for the 3-coil system from equation (14) orequation (15). The system parameters can then be used to optimize theoperation of the EV wireless charger.

FIG. 8 is the block diagram of a 2-coil EV wireless charging system withac/dc and dc/dc converters, and with unknown system parameters. Thischarging system does not use feedback from the receiver for controlpurposes. The parameters of the wireless power transfer system exceptM₁₂ and R_(L) (or P_(L)) are presumed to be known, i.e., all coilresistance, inductance and capacitance are given.

Once the system is initialized, based either on input voltage andcurrent, or input power, the System Parameters Identification unit 70provides the parameters M₁₂ and R_(L) (or P_(L)) to the WPTS Model 66,which determines the system outputs variables of U_(out), I_(out),P_(out) and η as in the Yin 2015 article. These variables are applied tocompensator/controller 72 along with a control objective signal thatdetermines the operating mode of the system. Controller 72 creates thesignals that drive the Frequency and Magnitude Variation circuit 74.Based on the output of circuit 74, system parameters are determined atdifferent frequencies or magnitudes in order to establish a sufficientnumber of equations to determine the unknown parameters.

If the active source is a dc source 60, an inverter 62 with an outputfilter is used to generate a sinusoidal voltage with controllablefrequency and magnitude for driving the first coil, i.e., TransmitterCoil 63. The selected frequencies and/or magnitudes for inverter 62 areset by circuit 74. Therefore, the input power is provided by energizingthe Transmitter Coil and such power will be wirelessly transmitted tothe last (Receiver) Coil 5 for powering the load 67. Since according tothe invention, measurements on the output load are to be eliminated;only the input voltage and the input current can be relied upon foroutput power control. Note that in the arrangement of FIG. 8 the outputload can be connected either in series with the last LC resonator or inparallel across the capacitor of the last LC resonator.

A sensor Block 64 detects the voltage and current outputs of theinverter 62, which drive the transmitter coil 63. The sensor Block 62outputs input voltage U_(in) and input current I_(in) to the systemparameter identification Block 70. The parameters generated by Block 70at particular frequencies and/or magnitudes are provided to thefrequency and magnitude varying circuit 74 as well as to the WPTS model66. Further, the input voltage U_(in) and input current I_(in) are alsoapplied to WPTS model 66.

The equations (4), (9) or (13) may be implemented and solved in systemparameter identification Block 70. Note that these equations require theinput voltage and input current only. Block 70 may be a microprocessorprogrammed to execute equation (4), (9) or (13) or some hardware deviceto perform the same function, such as a programmable gate array orapplication specific integrated circuit (ASIC).

In the operation of the circuit of FIG. 8, the Block 70 solves theequations with the (known) measured values of U_(in), I_(in) in order toobtain M₁₂ and R_(L) (or P_(L)). The calculations are made continuouslyat a high sampling rate (usually limited by the speed of the processor)to provide instantaneous output information for control and feedback.Such calculated values can be fed into any control scheme to meet thespecific control objectives of the wireless power transfer system. Basedon the chosen control scheme, e.g., control objective signal, the powerinverter 62 is operated so that it generates the appropriate sinusoidalvoltage at a controllable frequency and magnitude to meet the outputpower demand of the load 67 according to the control objective. Notethat load 67 receives the a.c. voltage from receiver coil 65 after ithas been ac/dc converted in circuit 76 and dc/dc converted in circuit77.

FIG. 9 is the block diagram of a 3-coil EV wireless charging systemaccording to the present invention, i.e., unknown system parameters aredetermined without the use of feedback from the receiver coil forcontrol purposes. It is structurally the same as the diagram of FIG. 8,except that it includes a relay coil 69. Thus, it includes ac/dcconverter 76 and dc/dc converter 77. Further, it includes the featuresof the Zhong 2015 article 1, in particular the current stress is shiftedfrom the primary driving circuit to the relay resonator and a largerelay current is generated to maximize the magnetic coupling with thereceiver coil for efficient power transfer. The equation (6), (orequation (10), or equation (12), or equation (14), or equation (15)) maybe implemented and solved in system parameter identification Block 70 toobtain M₁₀, M₁₂, M₀₂ and R_(L) or P_(L). Also these equations requirethe input voltage and input current only.

For a 2-coil wireless charging system, Equation (4), or (9), or (13) canbe used to calculate the mutual-inductance M₁₂ and the load resistanceR_(L) (or load power P_(L)) to make the whole wireless charging systemtransparent. Therefore the input frequency and voltage can be directlycontrolled, and it can be ensured that the system will be operated atits optimal point. Therefore, the proposed method of this invention canalso be linked with the maximum-efficiency tracking method proposed inthe article, W. Zhong and S. Hui, “Maximum Energy Efficiency Trackingfor Wireless Power Transfer Systems”, IEEE Transactions on PowerElectronics, vol. 30, pp. 4025-34, 2015 (the “Zhong 2015 article 2”),which is incorporated herein in its entirety. For the 2-coil wirelesscharging system of FIG. 8 the model is quite simple as shown in equation(3), and it can be rewritten as equation (16).

$\begin{matrix}{\begin{bmatrix}{V_{s}(\omega)} \\0\end{bmatrix} = {\begin{bmatrix}{R_{1} + {j\; \omega \; L_{1}}} & {j\; \omega \; M_{12}} \\{j\; \omega \; M_{12}} & {R_{2} + {j\; \omega \; L_{2}} + R_{L}}\end{bmatrix}\begin{bmatrix}{I_{1}(\omega)} \\{I_{2}(\omega)}\end{bmatrix}}} & (16)\end{matrix}$

Assuming V_(S)(ω) =U, from equation (3), we can get:

$\begin{matrix}{{I_{1}(\omega)} = \frac{U\left( {R_{2} + R_{L} + {j\; \omega \; L_{2}}} \right)}{{\left( {R_{1} + {j\; \omega \; L_{1}}} \right)\left( {R_{2} + R_{L} + {j\; \omega \; L_{2}}} \right)} + {\omega^{2}M_{12}^{2}}}} & (17) \\{{I_{2}(\omega)} = \frac{{- j}\; \omega \; M_{12}U}{{\left( {R_{1} + {j\; \omega \; L_{1}}} \right)\left( {R_{2} + R_{L} + {j\; \omega \; L_{2}}} \right)} + {\omega^{2}M_{12}^{2}}}} & (18)\end{matrix}$

And further, we can get the input power, output power and systemefficiency as shown below:

$\begin{matrix}{P_{in} = \frac{U^{2}\begin{pmatrix}{{R_{1}\left( {\left( {R_{2} + R_{L}} \right)^{2} + {\omega^{2}L_{2}^{2}} + \frac{1}{\omega^{2}C_{2}^{2}} - \frac{2L_{2}}{C_{2}}} \right)} +} \\{\omega^{2}{M_{12}^{2}\left( {R_{2} + {\omega \; R_{L}}}\; \right)}}\end{pmatrix}}{\begin{matrix}{\left( {{\omega^{2}M_{12}^{2}} + {R_{1}R_{2}} + {R_{1}R_{L}} - {\left( {{\omega \; L_{1}} - \frac{1}{\omega \; C_{1}}} \right)\left( {{\omega \; L_{2}} - \frac{1}{\omega \; C_{2}}} \right)}} \right)^{2} +} \\\left( {{\left( {{\omega \; L_{1}} - \frac{1}{\omega \; C_{1}}} \right)\left( {R_{2} + R_{L}} \right)} + {\left( {{\omega \; L_{2}} - \frac{1}{\omega \; C_{2}}} \right)R_{1}}} \right)^{2}\end{matrix}}} & (19) \\{P_{out} = \frac{U^{2}\omega^{2}M_{12}^{2}R_{L}}{\begin{matrix}{\left( {{\omega^{2}M_{12}^{2}} + {R_{1}R_{2}} + {R_{1}R_{L}} - {\left( {{\omega \; L_{1}} - \frac{1}{\omega \; C_{1}}} \right)\left( {{\omega \; L_{2}} - \frac{1}{\omega \; C_{2}}} \right)}} \right)^{2} +} \\\left( {{\left( {{\omega \; L_{1}} - \frac{1}{\omega \; C_{1}}} \right)\left( {R_{2} + R_{L}} \right)} + {\left( {{\omega \; L_{2}} - \frac{1}{\omega \; C_{2}}} \right)R_{1}}} \right)^{2}\end{matrix}}} & (20) \\{\eta = \frac{\omega^{2}M_{12}^{2}R_{L}}{\begin{matrix}{{R_{1}\left( {\left( {R_{2} + R_{L}} \right)^{2} + {\omega^{2}\; L_{2}^{2}} + \frac{1}{{\omega \;}^{2}C_{2}^{2}} - \frac{2L_{2}}{C_{2}}} \right)} +} \\{\omega^{2}{M_{12}^{2}\left( {R_{2} + {\omega \; R_{L}}} \right)}}\end{matrix}}} & (21)\end{matrix}$

FIG. 10 is the block diagram of a 2-coil EV wireless charging systemwith the features of the Zhong 2015 article 2 included, and with ac/dcconverter 76 and dc/dc converter 77. It also includes unknown systemparameters which are determined without the use of feedback from thereceiver for control purposes according to the present invention. In thearrangement of FIG. 10, the system parameters identification circuit 70is replaced with a controller of maximum efficiency tracking circuit 80that operates as described in the Zhong 2015 article 2. In particular,the switching mode converter 76 in the receiver is operated to emulatethe optimal load. The tracking circuit 80 causes the system to followthe maximum energy efficiency operating points of the wireless powertransfer system by searching for the minimum input power operating pointfor a given output power. Since this search is carried out on thetransmitter side there is no requirement for any wireless communicationfeedback from the receiver side.

If the system parameters are assumed to be the same as the systemdiscussed with respect to equation (13) and the Zhong 2015 article 2,L₁=105.35 μH, L₂=105.67 μH, M=46.107 μH, C₁=12.61 nF, C₂=12.57 nF,R₁=R₂=0.2 Ohm, and U=400 V, operating frequency f=138 kHz (the resonantfrequency of transmitter and also of the receiver), then the curves ofP_(out) vs. R_(L) and E_(ff) vs. R_(L)are as shown in FIG. 11, and thecurves of E_(ff) and P_(in) vs. input voltage when there is 5000 W ofconstant output power are shown in FIG. 12. FIG. 12 shows that if theload power is controlled by the battery charging controller to constantpower (in FIG. 9, P_(out)=5000 W) properly, the input power can besimply measured and the input voltage can be simply tuned to find theminimum point of input power as in the Zhong 2015 article 2. That isexactly the optimal operating point; i.e., where the system efficiencyis highest and output power can fit the demand.

While the invention has been particularly shown and described withreference to preferred embodiments thereof, it will be understood bythose skilled in the art that various changes in form and details may bemade therein without departing from the spirit and scope of theinvention. Additionally, many modifications may be made to adapt aparticular situation to the teachings of claimed subject matter withoutdeparting from the central concept described herein. Therefore, it isintended that claimed subject matter not be limited to the particularexamples disclosed, but that such claimed subject matter may alsoinclude all implementations falling within the scope of the appendedclaims, and equivalents thereof.

What we claimed is:
 1. A fast method for determining mutual inductanceterms representing the misalignment and distance between at least atransmitter coil and a magnetically coupled receiver coil in a wirelesspower charging system having a transmitter that receives input voltageand current that generates an electromagnetic field in the transmittercoil, and a receiver with the receiver coil that receives power from theelectromagnetic field and uses it to charge a load battery, comprisingthe steps of: exciting the transmitter coil of the wireless powercharging system under different conditions and measuring the response inorder to provide enough equivalent circuit system equations to at leastmatch the number of unknowns in the system equations, so that theunknowns, including at least the mutual inductance terms and the loadimpedance or load power, can be calculated with the knowledge of theinput voltage and input current of the transmitter coil, but withoutfeedback from the receiver coil; solving the system equations; andoptimizing the wireless power charging system based on the solved systemequations.
 2. The method of claim 1 wherein the different conditions aredifferent frequencies at which the transmitter coil is excited.
 3. Themethod of claim 1 wherein the different conditions are different inputvoltages/currents.
 4. The method of claim 2 wherein the excitation ofthe transmitter coil is at several frequencies around the resonantfrequency of the tuned resonance of the transmitter coil and thereceiver coil for operation at or near the resonant frequency to causeoptimal or near-optimal energy efficiency.
 5. The method of claim 1wherein the wireless power charging system further includes at least onerelay coil between the transmitter coil and the receiver coil, saidrelay coil receiving electromagnetic field from the transmitter coil andsending it to the receiver coil.
 6. The method of claim 5 wherein theexcitation of the transmitter coil is at several frequencies around theresonant frequency of the tuned resonance of the transmitter coil, therelay coil and the receiver coil for operation at or near the resonantfrequency to cause optimal or near-optimal energy efficiency.
 7. Themethod of claim 2 wherein the minimum number of the frequencies is equalto or more than the number of unknowns, and the unknowns comprise atleast the mutual inductance terms between the coils and the load poweror load impedance.
 8. The method of claim 7 wherein the number of thefrequencies is larger than the minimum number required to match thenumber of equations with that of the unknowns; and further including thestep of using a least square algorithm to achieve good accuracy for theparameters.
 9. The method of claim 3 wherein the load is aconstant-resistance type or a constant-power type during short periodsof time, and different voltage/current magnitudes are used to excite thetransmitter in order to generate all the required number of equations orsome of the required number of equations.
 10. The method of claim 1wherein once the mutual inductance terms and the load impedance or loadpower are determined, the step of optimizing involves choosing a properoperating frequency and input voltage/current to cause the wirelesspower charging system to operate at a maximum efficiency point and tomatch the charging power demand.
 11. The method of claim 1 furtherincluding the step of operating the load in a constant power mode, andwherein once the mutual inductance terms and the load power aredetermined, the step of optimizing involves choosing an optimal ornear-optimal operating frequency, and then changing only the inputvoltage to find the maximum efficiency operating point, and causing thesystem to stay at the maximum efficiency operating point even if theload power slowly changes within a small range.
 12. A wireless powercharging system comprising: a transmitter that receives input voltageand current and generates an electromagnetic field in a transmittercoil; a receiver with a receiver coil that receives power from theelectromagnetic field and uses it to charge a load battery; a controllerthat excites the transmitter coil of the wireless power charging systemunder different conditions and measuring the response, said controllerexciting the transmitter coil a sufficient number of times to provideenough equivalent circuit system equations to at least match the numberof unknowns, including at least the mutual inductance terms between thecoils and the load impedance or load power in the system equations; aprocessor that uses the measurements with the knowledge of the inputvoltage and input current of the transmitter coil to determine themutual inductance terms representing the misalignment and distancebetween the transmitter coil and the receiver coil, said processorfurther solving the system equations and optimizing the wireless powercharging system based on the solved system equations by setting anoperating frequency, and the input voltage and current.
 13. The wirelesspower charging system of claim 12 wherein the controller excites thetransmitter coil at different frequencies.
 14. The wireless powercharging system of claim 12 wherein the controller excites thetransmitter coil at different input voltages/currents.
 15. The wirelesspower charging system of claim 13 wherein the excitation of thetransmitter coil is at several frequencies around the resonant frequencyof the tuned resonance of the transmitter coil and the receiver coil foroperation at or near the resonant frequency to cause optimal ornear-optimal energy efficiency.
 16. The wireless power charging systemof claim 12 wherein the wireless power charging system further includesat least one relay coil between the transmitter coil and the receivercoil, said relay coil receiving electromagnetic field from thetransmitter coil and sending it to the receiver coil.
 17. The wirelesspower charging system of claim 15 wherein the excitation of thetransmitter coil by the controller is at several frequencies around theresonant frequency of the tuned resonance of the transmitter coil, therelay coil and the receiver coil for operation at or near the resonantfrequency to cause optimal or near-optimal energy efficiency.
 18. Thewireless power charging system of claim 13 wherein the minimum number ofthe frequencies is equal to or more than the number of unknowns in thesystem equations, and the unknowns comprise at least the mutualinductance terms between the coils and the load power or load impedance.19. The wireless power charging system of claim 18 wherein the number ofthe frequencies is larger than the minimum number required to match thenumber of equations with that of the unknowns; and further including thestep of using a least square algorithm to achieve good accuracy for theparameters.
 20. The wireless power charging system of claim 16 furtherincluding a power inverter to drive the transmitter, and wherein currentstress is shifted from the transmitter to the relay coil and a largerelay current is generated to maximize the magnetic coupling with thereceiver coil for efficient power transfer.
 21. The wireless powercharging system of claim 12 further including a switching mode converterin the receiver that is operated to emulate an optimal load, and atracking circuit at the transmitter that causes the system to follow themaximum energy efficiency operating points of the system by searchingfor the minimum input power operating point for a given output power.